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Mathematical construction used in homotopy theory
mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional
Simplicial_set
Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts)
Simplicial_complex
pro-simplicial set is an inverse system of simplicial sets. A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets
Pro-simplicial_set
In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn
Fibration_of_simplicial_sets
Endofunctor on the category of simplicial sets
simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets
Subdivision_(simplicial_set)
Construction for categories
join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to construct
Join_(simplicial_sets)
Simplicial object in the category of topological spaces
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the
Simplicial_space
Endofunctor on the category of simplicial sets
mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable
Extension_(simplicial_set)
Mathematical category with weak equivalences, fibrations and cofibrations
spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R-modules
Model_category
Abstraction useful in the construction and triangulation of topological spaces
In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation
Delta_set
Mathematical object
combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking
Abstract_simplicial_complex
Concept in algebraic topology
skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set K ∗ {\displaystyle
N-skeleton
Branch of mathematics
homology of the simplicial set S ∗ X {\displaystyle S_{*}X} . Also, the geometric realization | ⋅ | {\displaystyle |\cdot |} of a simplicial set is a CW complex
Homotopy_theory
Gives a homomorphism from homotopy groups to homology groups
theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition. Rational Hurewicz theorem: Let X be a
Hurewicz_theorem
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if f , g : X →
Simplicial_homotopy
Mathematical group of the homotopy classes of loops in a topological space
complex can be defined as the set of homotopy classes of 1-simplices. The fundamental group of an arbitrary simplicial set X {\displaystyle X} are defined
Fundamental_group
Concept in algebraic topology
especially algebraic topology, a weak equivalence between simplicial sets is a map between simplicial sets that is invertible in some weak sense. Formally, it
Weak equivalence between simplicial sets
Weak_equivalence_between_simplicial_sets
Construction for simplicial sets
In higher category theory in mathematics, the opposite simplicial set (or dual simplicial set) is an operation extending the opposite category (or dual
Opposite_simplicial_set
Generalization of a category
Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these
Quasi-category
Map between simplicial sets with lifting property
part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of
Kan_fibration
Branch of mathematics
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Algebraic_topology
Continuous mappings can be approximated by ones that are piecewise simple
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by
Simplicial approximation theorem
Simplicial_approximation_theorem
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of
Simplicial_map
Simplicial set constructed from the objects and morphisms of a small category
small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space
Nerve_(category_theory)
Type of category in mathematics
topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse system of simplicial sets one may
Topos
Simplicial object in the category of simplicial sets
bisimplicial set is a simplicial object in the category of simplicial sets, which themselves are simplicial objects in the category of sets. Many concepts
Bisimplicial_set
A subdivision (also called refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original
Subdivision (simplicial complex)
Subdivision_(simplicial_complex)
functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on
Simplicial_presheaf
cubical set is a set-valued contravariant functor on the category of (various) n-cubes. Cubical sets have been considered as an alternative to simplicial sets
Cubical_set
either the category of sets or the category of simplicial sets to the category of manifolds. A simplicial manifold is a simplicial complex for which the
Simplicial_manifold
Construction for simplicial sets
a simplicial set A {\displaystyle A} define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:
Twisted diagonal (simplicial sets)
Twisted_diagonal_(simplicial_sets)
Construction for simplicial sets
simplicial sets is an operation taking two simplicial sets to construct another simplicial set. It is closely related to the join of simplicial sets and
Diamond_operation
the independent sets of the graph. Formally, the independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that
Independence_complex
Category enriched over the category of simplicial sets
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often
Simplicially enriched category
Simplicially_enriched_category
functors from a category C to a category D. Set, the category of (small) sets. sSet, the category of simplicial sets. "weak" instead of "strict" is given the
Glossary_of_category_theory
Computational problem in algebraic topology
fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more. An abstract simplicial complex (ASC) is family of sets that
Simplicial complex recognition problem
Simplicial_complex_recognition_problem
Mathematical concept in topology
theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object
Simplicial_group
Topics referred to by the same term
Simplicially enriched category, a category enriched over the category of simplicial sets Simplicial object in the category of categories This disambiguation page
Simplicial_category
Concept in homotopy theory
all projective. The category SSet {\displaystyle {\textbf {SSet}}} of simplicial setspg 1.3 there is a model category structure where the fibrations are
Cofibration
In mathematics, a dendroidal set is a generalization of simplicial sets introduced by Moerdijk & Weiss (2007). They have the same relation to (colored
Dendroidal_set
Contravariant functor to Set
{V} } as a V {\displaystyle \mathbf {V} } -valued presheaf. A simplicial set is a Set-valued presheaf on the simplex category C = Δ {\displaystyle C=\Delta
Presheaf_(category_theory)
Category of non-empty finite ordinals and order-preserving maps
inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on Δ {\displaystyle \Delta
Simplex_category
Complex recording the pattern of intersections between a topological family's sets
product. This is the Čech nerve. By taking connected components we get a simplicial set, which we can realise topologically: | S ( π 0 ( C ) ) | {\displaystyle
Nerve_complex
if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to
Hypercovering
Analogue of homotopy type for algebraic varieties
U, n ≥ 0 {\displaystyle n\geq 0} ) by a single point. This gives a simplicial set which captures some information related to X and the étale topology
Étale_homotopy_type
Topological data
integrated usage of their mathematical properties. Consider any simplicial complex is a set composed of points (0 dimensions), line segments (1 dimension)
Simplex_tree
that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the
Density theorem (category theory)
Density_theorem_(category_theory)
Generalization of category theory
categories for any k. Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look
Higher_category_theory
Topics referred to by the same term
an operation combining two categories Join (simplicial sets), an operation combining two simplicial sets Join (sigma algebra), a refinement of sigma algebras
Join
Model structure on the category of simplicial sets
model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called fibrations, cofibrations and
Joyal_model_structure
the set of it and its neighbours is the union of two cliques, and is k-simplicial if the set is the union of k cliques. A vertex is co-simplicial if its
Simplicial_vertex
Area of mathematics
Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing
Discrete differential geometry
Discrete_differential_geometry
Mathematics glossary
definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations
Glossary of algebraic topology
Glossary_of_algebraic_topology
Branch of mathematics
particular, the category of simplicial rings is simplicially enriched, meaning the hom-sets are themselves simplicial sets. Also, there is a canonical
Derived_algebraic_geometry
Topics referred to by the same term
the set of tuples of values that satisfy the predicate Extension (semantics), the set of things to which a property applies Extension (simplicial set) Extension
Extension
Multi-dimensional generalization of triangle
optimization method with inequality constraints Simplicial complex Simplicial homology Simplicial set Space frame Spectrahedron Ternary plot Elte, E.L
Simplex
Equivalence between the categories of chain complexes and simplicial abelian groups
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there
Dold–Kan_correspondence
Topics referred to by the same term
value or E(X) EX (calculator key), to enter powers of 10 Extension (simplicial set), or Ex functor Extinct or EX, a conservation status ex, an author citation
EX
Model structure on the category of simplicial sets
model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called fibrations, cofibrations and
Kan–Quillen_model_structure
Abstract simplicial complex describing a graph's cliques
is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the
Clique_complex
Any collection of sets, or subsets of a set
any set in the family also belong to the family forms an abstract simplicial complex. An incidence structure consists of a set of points, a set of lines
Family_of_sets
Topological space formed from distances
way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and
Vietoris–Rips_complex
Formalism in general relativity
In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation
Regge_calculus
∞-category using Θ {\displaystyle \Theta } -sets = presheaves on Θ {\displaystyle \Theta } instead of simplicial sets = presheaves on Δ {\displaystyle \Delta
Joyal's_theta_category
Topics referred to by the same term
whereby replacing the edges by paths Subdivision (simplicial complex) Subdivision (simplicial set) Subdivision surface, in computer graphics Subdivision
Subdivision
are special model structures on slice categories of the category of simplicial sets. On them, postcomposition and pullbacks (due to its application in
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Abstract homotopical model for topological spaces
model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization
∞-groupoid
Mathematical category formed by reversing morphisms
using the opposite simplicial set. An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order
Opposite_category
Concept in algebraic topology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of
Simplicial_homology
History of maths
low-dimensional topology; Categorical logic and set theory in the categorical context such as algebraic set theory; Foundations of mathematics building on
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Largest open subset of some given set
notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior
Interior_(topology)
Establishes the concept of stabilization of homotopy groups
X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map X → Ω ( Σ X ) {\displaystyle X\to \Omega (\Sigma X)} induces
Freudenthal suspension theorem
Freudenthal_suspension_theorem
Topics referred to by the same term
See: Geometric realization of an abstract simplicial complex; Geometric realization of a simplicial set. This disambiguation page lists mathematics
Geometric_realization
Polish-American mathematician (1913–1998)
Category theory X-machine Weak dimension Projective module Shuffle algebra Simplicial set Standard complex Eilenberg's obstruction theory Eilenberg swindle Eilenberg–Ganea
Samuel_Eilenberg
Application of homotopy to algebraic varieties
a morphism of simplicial sheaves. We say that: f is a weak equivalence if, for any fibre functor x of T, the morphism of simplicial sets x ∗ f : x ∗ X
A¹_homotopy_theory
All points in the topological closure not belonging to the interior
idempotence. In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary
Boundary_(topology)
{\displaystyle {\mathcal {I}}} ist the category assigned to a small well-ordered set with initial element and if C {\displaystyle {\mathcal {C}}} has all small
Injective and projective model structure
Injective_and_projective_model_structure
Representation of mathematical space
topological properties of simplicial complexes and their generalizations, cell-complexes. An abstract simplicial complex above a set V {\displaystyle V} is
Triangulation_(topology)
Mathematics text
∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development
Higher_Topos_Theory
algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group Σ n {\displaystyle
Symmetric_spectrum
Research field in deep learning
but not limited to, graphs, simplicial complexes, cell complexes, combinatorial complexes and hypergraphs. Given a finite set S of abstract entities, a
Topological_deep_learning
Higher categorical generalization of a topos
homotopical model for topological spaces Simplicial set Kan complex – Concept in the theory of simplicial sets. Lurie 2009, Definition 6.1.0.4. Lurie 2009
∞-topos
Branch of topology
topology, since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined
General_topology
Mathematician, prolific contributor to homotopy theory
mid-1950s he made distinguished contributions to the theory of simplicial sets and simplicial methods in topology in general. In recognition of this, the
Daniel_Kan
Category where every morphism is invertible; generalization of a group
\hom _{\mathbf {sSet} }(X,N(G))} Here, π 1 ( X ) {\displaystyle \pi _{1}(X)} denotes the fundamental groupoid of the simplicial set X {\displaystyle
Groupoid
Topics referred to by the same term
loan in California and Nevada, US Simplicial link, a set of simplices "surrounding" a given vertex in a simplicial complex Link (knot theory), a collection
Link
Branch of geometry that studies combinatorial properties and constructive methods
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Discrete_geometry
Describes the fundamental group in terms of a cover by two open path-connected subspaces
Simplicial Sets and Van Kampen's Theorem (Discusses generalized versions of Van Kampen's theorem applied to topological spaces and simplicial sets).
Seifert–Van_Kampen_theorem
representation and processing, in geometrical modeling, they are related to simplicial set and to combinatorial topology, and this is a boundary representation
Generalized_map
Branch of mathematics
topological data analysis is to: Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological
Topology
Hypothesis in mathematical category theory
as a simplicial set satisfying the weak Kan condition, as done commonly today, then ∞-groupoids amounts exactly to Kan complexes (= simplicial sets with
Homotopy_hypothesis
lemma Polytope Simplex Simplicial complex CW complex Manifold Triangulation Barycentric subdivision Sperner's lemma Simplicial approximation theorem Nerve
List of general topology topics
List_of_general_topology_topics
Algebraic structure
structures. The E1-page has explicit terms with a differential coming from a simplicial set. Any smooth variety X admits a smooth compactification with complement
Hodge_structure
Mathematical concept
a univalent model of the Martin-Löf type theory with values in Kan simplicial sets can be found in a paper by Chris Kapulkin, Peter LeFanu Lumsdaine and
Univalent_foundations
topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex
Poset_topology
Concept in mathematical category theory
concept of hyperdoctrine. The category of elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology. More
Category_of_elements
This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive
Symmetric_product_(topology)
Canadian mathematician
proving the existence of a Quillen model structure on the category of simplicial sets whose weak equivalences generalize both equivalence of categories and
André_Joyal
Generalization of category
nerve N h c ( C ) {\displaystyle N^{hc}(C)} of a 2-category C is a simplicial set where each n-simplex is determined by the following data: n objects
2-category
Concepts in algebraic topology
nerve of this category and |-| is the topological realization of this simplicial set. Similarly, one can define a colimit as the left adjoint to the diagonal
Homotopy_colimit_and_limit
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